How Much can You Win when Your Adversary is Handicapped?

Abstract

We consider infinite games where a gambler plays a coin-tossing game against an adversary. The gambler puts stakes on heads or tails, and the adversary tosses a fair coin, but has to choose his outcome according to a previously given law known to the gambler. In other words, the adversary is not allowed to play all infinite heads-tails-sequences, but only a certain subset F of them.

We present an algorithm for the player which, depending on the structure of the set F, guarantees an optimal exponent of increase of the player’s capital, independently on which one of the allowed heads-tails-sequences the adversary chooses.

Using the known upper bound on the exponent provided by the maximum Kolmogorov complexity of sequences in F we show the optimality of our result.

A.K. Zvonkin and L.A. Levin, “Complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms”, Russian Math. Surveys, 25, 1970, 83 —124.Google Scholar